Question

How do you write with fractional exponents $\sqrt{3\left( {{x}^{2}} \right)}$?

Answer 1

Hint: Here in this question we have to express the given expression in terms of fractional exponents. We will convert the square root into fractional power and then by simplifying the obtained expression we will get the desired answer.

Complete step by step solution:
We have been given an expression $\sqrt{3\left( {{x}^{2}} \right)}$.
We have to write the given expression in the form of fractional exponents.
Now, first let us convert the radical form into fractional exponent. Now, we know that fractional exponents are the form of expressing the powers and roots together. Fractional exponent is also known as rational exponents.
Now, we know that $\sqrt{a}={{\left( a \right)}^{\dfrac{1}{2}}}$
So by applying the above formula to the given expression we will get
$\begin{align}
  & \Rightarrow \sqrt{3\left( {{x}^{2}} \right)} \\
 & \Rightarrow {{\left( 3{{x}^{2}} \right)}^{\dfrac{1}{2}}} \\
\end{align}$
Now, we know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow \left( {{3}^{\dfrac{1}{2}}}{{x}^{2{{\times }^{\dfrac{1}{2}}}}} \right) \\
 & \Rightarrow {{3}^{\dfrac{1}{2}}}.x \\
\end{align}$
Hence above is the required fractional exponent form of the given expression.

Note: To solve such types of questions students must know how to convert the radicals into fractional exponents. The possibility of mistake is while writing the exponent when we open the bracket. The point to be noted is that the exponent is to be written on both 3 and x in this particular question. If the number inside the square root is the perfect square then it will be easy to solve the powers.